Abstract
Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of $d^3$ vectors in $d$-dimensional Hilbert space that can be partitioned in two different ways: into $d$ SICs and into $d^2$ orthonormal bases. While a priori their existence may appear unlikely when $d>2$, we surprisingly answer it in the positive through an explicit construction for $d=4$. Remarkably this SIC-compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC-compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalisation of the six-state protocol.
Highlights
Quantum information theory has established a permanent link between the foundations of quantum theory and quantum information technologies
We show that SIC compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalization of the six-state protocol
We have introduced SIC compounds as an elegant and sophisticated discrete structure in Hilbert space
Summary
Quantum information theory has established a permanent link between the foundations of quantum theory and quantum information technologies This has reinvigorated interest in understanding the ultimate limitations of quantum states and measurements as discrete structures in Hilbert space. SICs have been studied in the context of quantum nonlocality [24,31,32,33] and they have an interesting foundational role in QBism [34] All this has triggered much interest in addressing the existence of SICs in general Hilbert space dimensions. We introduce a natural discrete Hilbert space structure that is compounded of many separate SICs. The resulting SIC compound is a set of d3 pure d-dimensional quantum states, denoted {|ψ jk } jk for j ∈ [d2] and k ∈ [d] We place the SIC compound at the heart of protocols for QKD, analyze their security under coherent attacks, and show their improved robustness as compared to the four-dimensional counterpart of the six-state protocol [39] (which extends the celebrated BB84 protocol [40])
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